OFFSET
0,2
COMMENTS
A295470, defined by A295470(n) = (20*n)!*(6*n)!*n! / ((12*n)!*(10*n)!*(3*n)!*(2*n)!), is one of the 52 sporadic integral factorial ratio sequences of height 1 found by V. I. Vasyunin (see Bober, Table 2, Entry 40). Here we are essentially considering the sequence {A295470(n/2) : n >= 0}. Fractional factorials are defined in terms of the gamma function; for example, (3*n/2)! := Gamma(1 + 3*n/2).
This sequence is only conjecturally an integer sequence.
Conjecture: the supercongruences a(n*p^r) == a(n*p^(r-1)) (mod p^(3*r)) hold for all primes p >= 5 and all positive integers n and r.
LINKS
J. W. Bober, Factorial ratios, hypergeometric series, and a family of step functions, arXiv:0709.1977 [math.NT], 2007; J. London Math. Soc., 79, Issue 2, (2009), 422-444.
FORMULA
a(n) ~ c^n * 1/sqrt(6*Pi*n), where c = (10/3)^5 * sqrt(3).
a(n) = 1600*(10*n - 1)*(10*n - 3)*(10*n - 7)*(10*n - 9)*(10*n - 11)*(10*n - 13)*(10*n - 17)*(10*n - 19)/(27*n*(n - 1)*(3*n - 2)*(3*n - 4)*(6*n - 1)*(6*n - 5)*(6*n - 7)*(6*n - 11))*a(n-2) with a(0) = 1 and a(1) = 168.
MAPLE
seq( simplify((10*n)!*(3*n)!*(n/2)!/((6*n)!*(5*n)!*(3*n/2)!*n!)), n = 0..15);
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Peter Bala, Jul 13 2023
STATUS
approved